Almost all almost regular c-partite tournaments with cgeq5 are vertex pancyclic

A tournament is an orientation of a complete graph and a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If D is a digraph, then let d + (x) be the outdgree and d ? (x) the indegree of the vertex x in D. The minimum (maximum) out-degree and the minimum (maximum) indegree of D are denoted by + ((+) and ? ((?), respectively. In addition, we deene = minf + ; ? g and = maxf + ; ? g. A digraph is regular when = and almost regular when ? 1. Very recently, the third author proved that all regular c-partite tournaments are vertex pancyclic when c 5, and that all, except possibly a nite number, regular 4-partite tournaments are vertex pancyclic. Clearly, in a regular multipartite tournament, each partite set has the same cardinality. As a supplement of Yeo's result we prove rst that an almost regular c-partite tournament with c 5 is vertex pancyclic, if all partite sets have the same cardinality. Second, we show that all almost regular c-partite tournaments are vertex pancyclic when c 8, and third that all, except possibly a nite number, almost regular c-partite tournaments are vertex pancyclic when c 5. 1. Terminology and introduction In this paper all digraphs are nite without loops and multiple arcs. A digraph without cycles of length two is an oriented graph. A c-partite or multipartite tournament is an orientation of a complete c-partite graph. A tournament is a c-partite tournament with exactly c vertices. The vertex set and the arc set of a digraph D are denoted by V (D) and E(D), respectively. If xy is an arc of a digraph D, then we say that x dominates y, and if A and B are two disjoint subsets of V (D) such that every vertex of A dominates every vertex of B, then we say that A dominates B, denoted by A ! B. If A and B are two disjoint subsets of a multipartite tournament D such that there is no arc from B to